Moduli Space Of Rational Curves Research Articles

Multimatroids and Rational Curves with Cyclic Action

Abstract We study the connection between multimatroids and moduli spaces of rational curves with cyclic action. Multimatroids are generalizations of matroids and delta-matroids that naturally arise in topological graph theory. The perspective of moduli of curves provides a tropical framework for studying multimatroids, generalizing the previous connection between type-$A$ permutohedral varieties (Losev–Manin moduli spaces) and matroids, and the connection between type-$B$ permutohedral varieties and delta-matroids. Specifically, we equate a combinatorial nef cone of the moduli space with the space of ${\mathbb {R}}$-multimatroids, a generalization of multimatroids, and we introduce the independence polytopal complex of a multimatroid, whose volume is identified with an intersection number on the moduli space. As an application, we give a combinatorial formula for a natural class of intersection numbers on the moduli space by relating to the volumes of independence polytopal complexes of multimatroids.

Free rational curves on low degree hypersurfaces and the circle method

We use a function field version of the Hardy-Littlewood circle method to study the locus of free rational curves on an arbitrary smooth projective hypersurface of sufficiently low degree. On the one hand this allows us to bound the dimension of the singular locus of the moduli space of rational curves on such hypersurfaces and, on the other hand, it sheds light on Peyre's reformulation of the Batyrev-Manin conjecture in terms of slopes with respect to the tangent bundle.

Rational curves on del Pezzo surfaces in positive characteristic

We study the space of rational curves on del Pezzo surfaces in positive characteristic. For most primes p p we prove the irreducibility of the moduli space of rational curves of a given nef class, extending results of Testa in characteristic 0 0 . We also investigate the principles of Geometric Manin’s Conjecture for weak del Pezzo surfaces. In the course of this investigation, we give examples of weak del Pezzo surfaces defined over F 2 ( t ) \mathbb F_2(t) or F 3 ( t ) \mathbb {F}_{3}(t) such that the exceptional sets in Manin’s Conjecture are Zariski dense.

Wonderful compactifications and rational curves with cyclic action

Abstract We prove that the moduli space of rational curves with cyclic action, constructed in our previous work, is realizable as a wonderful compactification of the complement of a hyperplane arrangement in a product of projective spaces. By proving a general result on such wonderful compactifications, we conclude that this moduli space is Chow-equivalent to an explicit toric variety (whose fan can be understood as a tropical version of the moduli space), from which a computation of its Chow ring follows.

Permutohedral complexes and rational curves with cyclic action

We define a moduli space of rational curves with finite-order automorphism and weighted orbits, and we prove that the combinatorics of its boundary strata are encoded by a particular polytopal complex that also captures the algebraic structure of a complex reflection group acting on the moduli space. This generalizes the situation for Losev-Manin's moduli space of curves (whose boundary strata are encoded by the permutohedron and related to the symmetric group) as well as the situation for Batyrev-Blume's moduli space of curves with involution, and it extends that work beyond the toric context.

Moduli spaces of rational curves on Fano threefolds

We prove several classification results for the components of the moduli space of rational curves on a smooth Fano threefold. In particular, we prove a conjecture of Batyrev on the growth of the number of components as the degree increases. The key to our approach is Geometric Manin's Conjecture which predicts the number of components parameterizing free curves.

The spaces of rational curves on del Pezzo threefolds of degree one

We prove the irreducibility of moduli spaces of rational curves on a general del Pezzo threefold of Picard rank 1 and degree 1. As corollaries, we confirm Geometric Manin’s conjecture and enumerativity of certain Gromov–Witten invariants for these threefolds.

Rational curves on prime Fano threefolds of index 1

We study the moduli spaces of rational curves on prime Fano threefolds of index 1. For general threefolds of most genera we compute the dimension and the number of irreducible components of these moduli spaces. Our results confirm Geometric Manin’s Conjecture in these examples and show the enumerativity of certain Gromov-Witten invariants.

Geometric Manin’s conjecture and rational curves

Let$X$be a smooth projective Fano variety over the complex numbers. We study the moduli space of rational curves on$X$using the perspective of Manin’s conjecture. In particular, we bound the dimension and number of components of spaces of rational curves on$X$. We propose a geometric Manin’s conjecture predicting the growth rate of a counting function associated to the irreducible components of these moduli spaces.

Genus two enumerative invariants in del-Pezzo surfaces with a fixed complex structure

We obtain a formula for the number of genus two curves with a fixed complex structure of a given degree on a del-Pezzo surface that pass through an appropriate number of generic points of the surface. This is done by extending the symplectic approach of Aleksey Zinger. This enumerative problem is expressed as the difference between the symplectic invariant and an intersection number on the moduli space of rational curves on the surface.

Geometry of moduli spaces of rational curves in linear sections of Grassmannian Gr(2,5)

We prove that the moduli spaces of rational curves of degree at most 3 in linear sections of the Grassmannian Gr(2,5) are all rational varieties. We also study their compactifications and birational geometry.

Extremal divisors on moduli spaces of rational curves with marked points

We study effective divisors on $\overline{M}_{0,n}$, focusing on hypertree divisors introduced by Castravet and Tevelev and the proper transforms of divisors on $\overline{M}_{1,n-2}$ introduced by Chen and Coskun. Results include a database of hypertree divisor classes and closed formulas for Chen--Coskun divisor classes. We relate these two types of divisors, and from this construct extremal divisors on $\overline{M}_{0,n}$ for $n \geq 7$ that furnish counterexamples to the conjectural description of the effective cone of $\overline{M}_{0,n}$ given by Castravet and Tevelev.

Genus one enumerative invariants in del-Pezzo surfaces with a fixed complex structure

We obtain a formula for the number of genus one curves with a fixed complex structure of a given degree on a del-Pezzo surface that pass through an appropriate number of generic points of the surface. This enumerative problem is expressed as the difference between the symplectic invariant and an intersection number on the moduli space of rational curves.

Rational curves and special metrics on twistor spaces

A Hermitian metric $\omega$ on a complex manifold is called SKT or pluriclosed if $dd^c\omega=0$. Let M be a twistor space of a compact, anti-selfdual Riemannian manifold, admitting a pluriclosed Hermitian metric. We prove that in this case M is K\"ahler, hence isomorphic to $\C P^3$ or a flag space. This result is obtained from rational connectedness of the twistor space, due to F. Campana. As an aside, we prove that the moduli space of rational curves on the twistor space of a K3 surface is Stein.

Complex Static Skew-Symmetric Output Feedback Control

We study the problem of feedback control for skew-symmetric and skew-Hamiltonian transfer functions using skew-symmetric controllers. This extends work of Helmke et al., who studied static symmetric feedback control of symmetric and Hamiltonian linear systems. We identify spaces of linear systems with symmetry as natural subvarieties of the moduli space of rational curves in a Grassmannian, give necessary and sufficient conditions for pole placement by static skew-symmetric complex feedback, and use Schubert calculus for the orthogonal Grassmannian to count the number of complex feedback laws when there are finitely many of them. Finally, we also construct a real skew-symmetric linear system with only real feedback for any set of real poles.

Compactified moduli spaces of rational curves in projective homogeneous varieties

The space of smooth rational curves of degree $d$ in a projective variety $X$ has compactifications by taking closures in the Hilbert scheme, the moduli space of stable sheaves or the moduli space of stable maps respectively. In this paper we compare these compactifications by explicit blow-ups and -downs when $X$ is a projective homogeneous variety and $d\leq 3$. Using the comparison result, we calculate the Betti numbers of the compactifications when $X$ is a Grassmannian variety.

Contact homology of Hamiltonian mapping tori

In the general geometric setup for symplectic field theory the contact manifolds can be replaced by mapping tori M_Φ of symplectic manifolds (M,ω) with symplectomorphisms Φ . While the cylindrical contact homology of M_Φ is given by the Floer homologies of powers of Φ , the other algebraic invariants of symplectic field theory for M_Φ provide natural generalizations of symplectic Floer homology. For symplectically aspherical M and Hamiltonian Φ we study the moduli spaces of rational curves and prove a transversality result, which does not need the polyfold theory by Hofer, Wysocki and Zehnder. We use our result to compute the full contact homology of M_Φ \cong S^1 × M .

Notes on Certain (0,2) Correlation Functions

In this paper we shall describe some correlation function computations in perturbative heterotic strings that, for example, in certain circumstances can lend themselves to a heterotic generalization of quantum cohomology calculations. Ordinary quantum chiral rings reflect worldsheet instanton corrections to correlation functions involving products of Dolbeault cohomology groups on the target space. The heterotic generalization described here involves computing worldsheet instanton corrections to correlation functions defined by products of elements of sheaf cohomology groups. One must not only compactify moduli spaces of rational curves, but also extend a sheaf (determined by the gauge bundle) over the compactification, and linear sigma models provide natural mechanisms for doing both. Euler classes of obstruction bundles generalize to this language in an interesting way.

RATIONAL FAMILIES OF VECTOR BUNDLES ON CURVES

Let C be a smooth projective complex curve of genus g≥2 and let M be the moduli space of rank 2, stable vector bundles on C, with fixed determinant of degree 1. For any k≥1, we find all the irreducible components of the space of rational curves on M, of degree k. In particular, we find the maximal rationally connected fibrations of these components. We prove that there is a one-to-one correspondence between moduli spaces of rational curves on M and moduli spaces of rank 2 vector bundles on ℙ1×C.

Hyper-Kähler Hierarchies and Their Twistor Theory

A twistor construction of the hierarchy associated with the hyper-K\"ahler equations on a metric (the anti-self-dual Einstein vacuum equations, ASDVE, in four dimensions) is given. The recursion operator R is constructed and used to build an infinite-dimensional symmetry algebra and in particular higher flows for the hyper-K\"ahler equations. It is shown that R acts on the twistor data by multiplication with a rational function. The structures are illustrated by the example of the Sparling-Tod (Eguchi-Hansen) solution. An extended space-time ${\cal N}$ is constructed whose extra dimensions correspond to higher flows of the hierarchy. It is shown that ${\cal N}$ is a moduli space of rational curves with normal bundle ${\cal O}(n)\oplus{\cal O}(n)$ in twistor space and is canonically equipped with a Lax distribution for ASDVE hierarchies. The space ${\cal N}$ is shown to be foliated by four dimensional hyper-K{\"a}hler slices. The Lagrangian, Hamiltonian and bi-Hamiltonian formulations of the ASDVE in the form of the heavenly equations are given. The symplectic form on the moduli space of solutions to heavenly equations is derived, and is shown to be compatible with the recursion operator.